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%I #19 Aug 10 2018 11:19:00
%S 1,1,8,91,1474,30765,785053,23682833,824522797,32537599175,
%T 1435199414014,69973425937141,3736662443907962,216901789032691605,
%U 13598124265965160130,915670842666691879191,65913110467411283181409,5050836914009172555862713,410501468976427335127369669,35269929119728622895198302033,3194195105084750546987502710855
%N G.f.: Sum_{n>=0} (2 + (1+x)^n)^n / (3 + (1+x)^n)^(n+1).
%C The following identity holds for |y| <= 1 and fixed real k > 0:
%C Sum_{n>=0} (k + y^n)^n/(1+k + y^n)^(n+1) = Sum_{n>=0} (y^n - 1)^n/(1+k - k*y^n)^(n+1).
%H Paul D. Hanna, <a href="/A302614/b302614.txt">Table of n, a(n) for n = 0..200</a>
%F G.f.: Sum_{n>=0} ((1+x)^n - 1)^n / (3 - 2*(1+x)^n)^(n+1).
%F a(n) ~ c * d^n * n! / sqrt(n), where d = 4.64471605501103711823541367464... and c = 0.270134222044915506270113032... - _Vaclav Kotesovec_, Aug 10 2018
%e G.f.: A(x) = 1 + x + 8*x^2 + 91*x^3 + 1474*x^4 + 30765*x^5 + 785053*x^6 + 23682833*x^7 + 824522797*x^8 + 32537599175*x^9 + ...
%e such that
%e A(x) = 1/4 + (2 + (1+x))/(3 + (1+x))^2 + (2 + (1+x)^2)^2/(3 + (1+x)^2)^3 + (2 + (1+x)^3)^3/(3 + (1+x)^3)^4 + (2 + (1+x)^4)^4/(3 + (1+x)^4)^5 + (2 + (1+x)^5)^5/(3 + (1+x)^5)^6 + (2 + (1+x)^6)^6/(3 + (1+x)^6)^7 + ...
%e Also,
%e A(x) = 1 + ((1+x) - 1)/(3 - 2*(1+x))^2 + ((1+x)^2 - 1)^2/(3 - 2*(1+x)^2)^3 + ((1+x)^3 - 1)^3/(3 - 2*(1+x)^3)^4 + ((1+x)^4 - 1)^4/(3 - 2*(1+x)^4)^5 + ((1+x)^5 - 1)^5/(3 - 2*(1+x)^5)^6 + ((1+x)^6 - 1)^6/(3 - 2*(1+x)^6)^7 + ...
%e RELATED INFINITE SERIES.
%e (1) At x = -1/3: the following sums are equal
%e S1 = Sum_{n>=0} 3^n * (2*3^n + 2^n)^n / (3^(n+1) + 2^n)^(n+1),
%e S1 = Sum_{n>=0} (-3)^n * (3^n - 2^n)^n / (3^(n+1) - 2^(n+1))^(n+1).
%e Explicitly,
%e S1 = 1/4 + 3*8/11^2 + 9*22^2/31^3 + 27*62^3/89^4 + 81*178^4/259^5 + 243*518^5/761^6 + 729*1522^6/2251^7 + 2187*4502^7/6689^8 + 6561*13378^8/19939^9 + 19683*39878^9/59561^10 + ...
%e S1 = 1 - 3*1/5^2 + 9*5^2/19^3 - 27*19^3/65^4 + 81*65^4/211^5 - 243*211^5/665^6 + 729*665^6/2059^7 - 2187*2059^7/6305^8 + 6561*6305^8/19171^9 - 19683*19171^9/58025^10 + ...
%e where S1 = 0.90501051059439877583104471171480036033530856741889530664913...
%e (2) At x = -1/2: the following sums are equal
%e S2 = Sum_{n>=0} 2^n * (2^(n+1) + 1)^n / (3*2^n + 1)^(n+1),
%e S2 = Sum_{n>=0} (-2)^n * (2^n - 1)^n / (3*2^n - 2)^(n+1).
%e Explicitly,
%e S2 = 1/4 + 2*5/7^2 + 4*9^2/13^3 + 8*17^3/25^4 + 16*33^4/49^5 + 32*65^5/97^6 + 64*129^6/193^7 + 128*257^7/385^8 + 256*513^8/769^9 + 512*1025^9/1537^10 + ...
%e S2 = 1 - 2*1/4^2 + 4*3^2/10^3 - 8*7^3/22^4 + 16*15^4/46^5 - 32*31^5/94^6 + 64*63^6/190^7 - 128*127^7/382^8 + 256*255^8/766^9 - 512*511^9/1534^10 + ...
%e where S2 = 0.90222608896798122564942421232120719521782835530371831680447...
%o (PARI) {a(n) = my(A=1, o=x*O(x^n)); A = sum(m=0, n, ((1+x +o)^m - 1)^m / (3 - 2*(1+x +o)^m)^(m+1)); polcoeff(A, n)}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A122400, A302598, A302615.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Apr 10 2018
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